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2003 Course Operations Research

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Total No. of Questions : 12] P1323 [Total No. of Pages : 4 [3764]-174 B.E. (Production) OPERATIONS RESEARCH (2003 Course) Time : 3 Hours] [Max. Marks : 100 Instructions to the candidate: 1) Solve one question from every unit in each section. 2) Answers to the two sections should be written in separate answer books. 3) Neat diagrams must be drawn wherever necessary. 4) Figures to the right indicate full marks. 5) Use of electronic pocket calculators is allowed. 6) Assume suitable data, if necessary. Q1) a) b) SECTION - I UNIT - I How do you recognise that an LP problem is unbounded while using the simplex method? [6] The Handy-Dandy company wishes to schedule the production of a kitchen appliance that requires two resources - labor and material. The company is considering three different models and its production engineering department has furnished the following data: Model A B C Labor (Hrs. per unit) 7 3 6 Material (Kgs per unit) 4 4 5 Profit (Rs. per unit) 4 2 3 The supply of raw material is restricted to 200 Kgs/day. The daily availability of labor is 150 Hrs. Formulate a linear programming model to determine maximum profit. [10] OR Q2) a) What are artificial variables? Why do you need them? How do they differ from slack/surplus variables? [6] P.T.O. 1 b) Q3) a) b) Q4) a) b) Solve the following by dual method. Minimise Z = x1 + 4x2 + 3x4 Subject to x1+ 2x2 - x3 + x4 3 -2x1 - x2 + 4x3 + x4 2 x1, x2, x3, x4 0. [10] UNIT - II What are the similarities and differences between a transportation problem and a transshipment problem? [6] We have three reservoirs with daily supplies of 15, 20 and 25 million liters of fresh water respectively. On each day we must supply four cities A, B, C, D whose demands are 8, 10, 12 and 15 respectively. The cost of pumping per million liters is given below in thousand Rupees. Cities ABCD 1 2 3 4 5 Reservoirs 2 3 2 5 2 3 4 1 2 3 Use the vogels Approximation method to determine the cheapest pumping schedule if excess water can be disposed of at no cost. [10] OR Explain why the transportation problem solving algorithm is not appropriate for solving the assignment problem? [6] Find the optimal assignment of fourjons and four machines when the cost of assignment is given by the following table : J1 J2 J3 J4 M1 10 9 8 7 3 4 5 6 M2 M3 2 1 1 2 4 3 5 6 [10] M4 Q5) a) UNIT - III What are the shadow prices and how are they computed? What is their relationship to the dual problem? [6] b) Solve the following integer problem by brands and bound algorithm. [3764]-174 2 -2- Maximize Z = 5x1 + 8x2 Subject to x1 + x2 6 5x1 + 9x2 45 x1 , x2 Q6) a) b) Q7) a) b) 0 and integer. [12] OR List 6 reasons why simulation analysis is more appropriate for many real world problems. [6] Define in words the following concepts in dynamic programming : i) stage, ii) state, iii) statevariable, iv) Transformation function, v) Return function, vi) Decision variable, vii) Translation variable. [12] SECTION - II UNIT - IV What is the difference between a goal and a constraint as used in goal programming? [6] The following mortality rates have been observed for a certain type of fuses: End of week 1 2 3 4 5 % Failing to Q8) a) b) Q9) a) b) 15 35 75 100 [10] There are 1000 fuses in use and it costs Rs. 5 to replace an individual fuse. If all the fuses were replaced simultaneousoly, it would cost Rs.1.25 per fuse. It is now proposed to replace all the fuses at fixed intervals of time irrespective of their state and to continue replacing burnt out fuses as they fail. At what intervals group replacement should be made? OR Explain the concept of geometric programming. [6] Compare the policies adopted in group replacement as against individual replacement. In what circumstances, still individual replacement will prevail the other? [10] UNIT - V Describe Kendall's notation for identifying queueing models with examples. [6] Consider a game having the following payoff matrix. Player B B1 B2 Player A A1 2 6 A2 -2 [3764]-174 3 5 -3- Show that what ever the value of max be, the game is strictly deterministric. ii) Solve for the value of the game. [10] OR Explain the following terms in game theory. [6] i) Saddle point ii) Min Max Principle iii) Value of game i) Q10)a) b) Q11)a) b) What is meant by queue discipline? Describe some of the performance measures used in analysing queues. [10] UNIT - VI What are the essential differences between PERT and CPM? Consider a project of eight jobs to be performed. Job [6] [12] Predecessors Normal Time Crash Time Cost of Crashing (days) (days) per day (Rs.) A 10 7 4 B 5 4 2 C B 3 2 2 D A, C 4 3 3 E A, C 5 3 3 F D 6 3 5 G E 5 2 1 H F, G 5 4 4 Given the overhead costs as Rs. 5 per day, determine the optimal duration of the project in terms of both the crashing and overhead costs and also develop an optimal project schedule. Q12)a) b) OR What is the meaning of critical path in project management? Why it is the longest path? [6] Explain the following terms in networks. [12] i) Event, ii) slack, iii) Free Float, iv) Independent float, v) Total float, vi) Beta distribution curve. [3764]-174 4 -4-

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